Integrated research of the general hypergeometric systems and nonlinear integrable systems

本研究的目的是1)一般超几何函数和Okubo系统的研究，2)包括painlevel方程在内的非线性可积系统的研究。GL（N，C）正则元的集中子的共轭类由N的划分确定，一般超几何函数是集中子的泛覆盖群的Radon变换得到的格拉斯曼流形Gr（N， N）上的函数。我们明确地确定了与一般超几何函数的积分表示相关的代数de Rham上同调群。在情况n=2和情况n>2中，这个问题已经解决，分区为（1，…，1），(n)。对于分区（q, 1，…，1），证明了上同调群的纯粹性，确定了上同调的维数，给出了上同调的显式基。这一结果对于构造表征一般超几何的高斯-曼宁系统具有重要意义。在划分为(N)的情况下，我们构造了de Rham上同调的交点理论，并用偏舒尔多项式表示了交点数。在这个计算中，我们认识到在奇点理论中扁平基的类似物起着重要的作用。对于P^1上无辅助参数的Schlesinger型微分方程，利用Okubo系统的相应结果证明了其解具有积分表示。这种积分表示是GKZ超几何函数的一种特殊情况。因此，如何理解GKZ超几何函数框架下的辅助参数自由方程，并将其推广到具有不规则奇异点的方程，是一个有趣的问题。对于Painleve方程，我们证明了每个Painleve方程的初始条件空间存在一个辛结构，并且证明了初始条件空间的几何形状决定了Painleve方程。我们发现了一个有趣的现象，即painlevel II的一系列有理解的生成函数与由Airy函数得到的函数在无穷远处的渐近展开式重合。少

The objective of this research is 1) the study of the general hypergeometric functions and Okubo systems, 2) the study of nonlinear integrable systems including Painleve equations. The conjugacy classes of the centralizers of regular elements of GL(N,C) are determined by partitions of N. The general hypergeometric functions are functions on the Grassmannian manifold Gr(n,N) obtained by the Radon transformation of characters of universal covering groups of centralizers. We explicitly determined the algebraic de Rham cohomology groups associated with the integral representation of the general hypergeometric functions. This problem has been isolved in the case n=2 and in the case n>2 with the partitions (1,…,1), (N). For the case of partitions (q, 1,…,1), we proved the purity of the cohomology group, determined the dimension of the top cohomology and gave an explicit basis for it. This result will be important in constructing the Gauss-Manin system characterizing the general hypergeometri … More c functions. In the case where the partition is (N), we constructed the intersection theory of de Rham cohomology and expressed the intersection numbers in terms of skew Schur polynomials. In this computation, we recognized that an analogue of flat basis plays an important roles which appears in the theory of singularity. For the differential equation of Schlesinger type on P^1 without accessory parameters, we showed that the solutions have integral representations using the corresponding result for Okubo system. This integral representation is a particular case of that of GKZ hypergeometric functions. Thus it may be an interesting problem to understand the accessory parameter free equations in the framework of GKZ hypergeometric functions and to generalize this problem to the equations with irregular singularities.For the Painleve equations, we showed that there is a symplectic structure for the space of initial conditions for each Painleve equation and also showed that the geometry of the space of initial conditions determines the Painleve equation. We found an interesting phenomenon that a generating function for a series of rational solutions of Painleve II coincides with the asymptotic expansion at infinity of the function obtained from Airy function. Less